Symbols and Notations
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Symbols and Notations
Basics
X; Y X1 ; X2 ; : : : ; Xp X D .X1 ; : : : ; Xp /> X A; B ; X;Y † 1n
Random variables or vectors Random variables Random vector X has distribution Matrices Matrices Data matrices Covariance matrix Vector of ones .1; : : : ; 1/> „ ƒ‚ …
0n
Vector of zeros .0; : : : ; 0/> „ ƒ‚ …
I.:/
Indicator function, i.e. for a set M is I D 1 on M , I D 0 otherwise p 1 Implication Equivalence Approximately equal Kronecker product if and only if, equivalence
53 60 81 80 54
n-times
54
n-times
i ) , ˝ iff
© Springer-Verlag Berlin Heidelberg 2015 W.K. Härdle, L. Simar, Applied Multivariate Statistical Analysis, DOI 10.1007/978-3-662-45171-7_21
557
558
21 Symbols and Notations
Mathematical Abbreviations
tr.A/ hull.x1 ; : : : ; xk / diag.A/ rank.A/ det.A/ C.A/
Trace of matrix A Convex hull of points fx1 ; : : : ; xk g Diagonal of matrix A Rank of matrix A Determinant of matrix A Column space of matrix A
Samples
x; y x1 ; : : : ; xn D fxi gniD1 X D fxij giD1;:::;nIj D1;:::;p x.1/ ; : : : ; x.n/ H
Observations of X and Y Sample of n observations of X (n p) data matrix of observations of X1 ; : : : ; Xp or of X D .X1 ; : : : ; Xp /T The order statistic of x1 ; : : : ; xn Centering matrix, H D In n1 1n 1> n
Densities and Distribution Functions
f .x/ f .x; y/ fX .x/; fY .y/ fX1 .x1 /; : : : ; fXp .x2 / fOh .x/ F .x/ F .x; y/ FX .x/; FY .y/ FX1 .x1 /; : : : ; fXp .xp / '.x/ ˆ.x/ 'X .t / mk j
Density of X Joint density of X and Y Marginal densities of X and Y Marginal densities of X1 ; : : : ; Xp Histogram or kernel estimator of f .x/ Distribution function of X Joint distribution function of X and Y Marginal distribution functions of X and Y Marginal distribution functions of X1 ; : : : ; Xp Density of the standard normal distribution Standard normal distribution function Characteristic function of X k-th moment of X Cumulants or semi-invariants of X
11
81 5 90
21 Symbols and Notations
559
Moments
E X; E Y XY D Cov.X; Y / XX D Var.X/ Cov.X; Y / XY D p Var.X/ Var.Y / †XY D Cov.X; Y / †XX D Var.X/
Mean values of random variables or vectors X and Y Covariance between random variables X and Y Variance of random variable X
80 80 80
Correlation between random variables X and Y
84
Covariance between random vectors X and Y , i.e., Cov.X; Y / D E.X EX/.Y EY /> Covariance matrix of the random vector X
Empirical Moments
xD sXY
n 1X xi n iD1 n 1X D .xi x/.yi y/ n iD1
n 1X .xi x/2 n iD1 sXY rXY D p sXX sY Y S D fsXi Xj g D x > Hx
sXX D
R D frXi Xj g D D 1=2 SD 1=2
Average of X sampled by fxi giD1;:::;n
7
Empirical covariance of random variables X and Y sampled by fxi giD1;:::;n and fyi giD1;:::;n
80
Empirical variance of random variable X sampled by fxi giD1;:::;n
80
Empirical correlation of X and Y
84
Empirical covariance matrix of X1 ; : : : ; Xp or of the random vector X D .X1 ; : : : ; Xp /> Empirical correlation matrix of X1 ; : : : ; Xp or of the random vector X D .X1 ; : : : ; Xp />
80, 90
Distributions
'.x/ ˆ.x/ N.0; 1/ N.; 2 / Np .
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